On substantial fractional difference operator

Abstract

AbstractIn this article, new definitions of fractional substantial sum and difference operators are introduced. Some properties are established and used to generate a fixed point operator for an arbitrary real order substantial system with conditions involving fractional order difference. We inspect solution existence and Ulam–Hyers–Rassias stability. A property missing in the literature for delta Laplace transform, namely delta exponential shift, is established and used for finding delta Laplace transform for the newly introduced substantial fractional sum and difference.